Metamath Proof Explorer


Theorem islln4

Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012)

Ref Expression
Hypotheses llnset.b 𝐵 = ( Base ‘ 𝐾 )
llnset.c 𝐶 = ( ⋖ ‘ 𝐾 )
llnset.a 𝐴 = ( Atoms ‘ 𝐾 )
llnset.n 𝑁 = ( LLines ‘ 𝐾 )
Assertion islln4 ( ( 𝐾𝐷𝑋𝐵 ) → ( 𝑋𝑁 ↔ ∃ 𝑝𝐴 𝑝 𝐶 𝑋 ) )

Proof

Step Hyp Ref Expression
1 llnset.b 𝐵 = ( Base ‘ 𝐾 )
2 llnset.c 𝐶 = ( ⋖ ‘ 𝐾 )
3 llnset.a 𝐴 = ( Atoms ‘ 𝐾 )
4 llnset.n 𝑁 = ( LLines ‘ 𝐾 )
5 1 2 3 4 islln ( 𝐾𝐷 → ( 𝑋𝑁 ↔ ( 𝑋𝐵 ∧ ∃ 𝑝𝐴 𝑝 𝐶 𝑋 ) ) )
6 5 baibd ( ( 𝐾𝐷𝑋𝐵 ) → ( 𝑋𝑁 ↔ ∃ 𝑝𝐴 𝑝 𝐶 𝑋 ) )